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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of $Sl(2,\mathbb R)$
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by Bertram Kostant PDF
Represent. Theory 4 (2000), 181-224 Request permission


Analogous to the holomorphic discrete series of $Sl(2,\mathbb R)$ there is a continuous family $\{\pi _r\}$, $-1<r<\infty$, of irreducible unitary representations of $G$, the simply-connected covering group of $Sl(2,\mathbb R)$. A construction of this series is given in this paper using classical function theory. For all $r$ the Hilbert space is $L_2((0,\infty ))$. First of all one exhibits a representation, $D_r$, of $\mathfrak g= \mathit {Lie} G$ by second order differential operators on $C^\infty ((0,\infty ))$. For $x\in (0,\infty )$, $-1<r<\infty$ and $n\in \mathbb Z_+$ let $\varphi _n^{(r)}(x)= e^{-x}x^{\frac {r}{2}}L_n^{(r)}(2x)$ where $L_n^{(r)}(x)$ is the Laguerre polynomial with parameters $\{n,r\}$. Let $\mathcal H_r^{HC}$ be the span of $\varphi _n^{(r)}$ for $n\in \mathbb Z_+$. Next one shows, using a famous result of E. Nelson, that $D_r|{\mathcal H}_r^{HC}$ exponentiates to the unitary representation $\pi _r$ of $G$. The power of Nelson’s theorem is exhibited here by the fact that if $0<r<1$, one has $D_r=D_{-r}$, whereas $\pi _r$ is inequivalent to $\pi _{-r}$. For $r=\frac 12$, the elements in the pair $\{\pi _{\frac {1}{2}},\pi _{-\frac {1}{2}}\}$ are the two components of the metaplectic representation. Using a result of G.H. Hardy one shows that the Hankel transform is given by $\pi _r(a)$ where $a\in G$ induces the non-trivial element of a Weyl group. As a consequence, continuity properties and enlarged domains of definition, of the Hankel transform follow from standard facts in representation theory. Also, if $J_r$ is the classical Bessel function, then for any $y\in (0,\infty )$, the function $J_{r,y}(x)=J_r(2\sqrt {xy})$ is a Whittaker vector. Other weight vectors are given and the highest weight vector is given by a limiting behavior at $0$.
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Additional Information
  • Bertram Kostant
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • Email:
  • Received by editor(s): December 2, 1999
  • Received by editor(s) in revised form: January 21, 2000
  • Published electronically: April 26, 2000
  • Additional Notes: Research supported in part by NSF grant DMS-9625941 and in part by the KG&G Foundation
  • © Copyright 2000 American Mathematical Society
  • Journal: Represent. Theory 4 (2000), 181-224
  • MSC (2000): Primary 22D10, 22E70, 33Cxx, 33C10, 33C45, 42C05, 43-xx, 43A65
  • DOI:
  • MathSciNet review: 1755901